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Abstract

Wide variety of engineering design problems can be formulated as constrained optimization problems where the shape and topology of the domain are optimized to reduce costs while satisfying certain constraints. Several mathematical approaches were developed to address the problem of finding optimal design of an engineered structure. Recent works [, ] have demonstrated the feasibility of boundary element method as a tool for topological-shape optimization. However, it was noted that the approach has certain drawbacks, and in particular high computational cost of the iterative optimization process. In this short note we suggest ways to address critical limitations of boundary element method as a tool for topological-shape optimization. We validate our approaches by supplementing the existing complex variables boundary element code for elastostatic problems with robust tools for the fast topological-shape optimization. The efficiency of the approach is illustrated with a numerical example.

Abstract

Topological optimization finds a material density distribution minimizing a functional of the solution of a partial differential equation (PDE), subject to a set of constraints (typically, a bound on the volume or mass of the material).

Using a finite elements discretization (FEM) of the PDE and functional we obtain an integer programming problem. Due to approximation error of the FEM discretization, optimization problem becomes mesh-depended and possess false, physically inadequate optimums, while functional value heavily depends on the fineness of discretization scheme used to compute it. To alleviate this problem, we propose regularization of given functional by error estimate of the FEM discretization. This regularization provides robustness of solutions and improves obtained functional values as well.

While the idea is broadly applicable, in this paper we apply our method to the heat conduction optimization. Problems of this type are of practical importance in design of heat conduction channels, heat sinks and other types of heat guides.

Abstract

In this paper, we first establish a kind of weighted space-time L2 estimate, which belongs to Keel–Smith–Sogge-type estimates, for perturbed linear elastic wave equations. This estimate refines the corresponding one established by the second author [D. Zha, Space-time L2 estimates for elastic waves and applications, J. Differential Equations 263 2017, 4, 1947–1965] and is proved by combining the methods in the former paper, the first author, Wang and Yokoyama’s paper [K. Hidano, C. Wang and K. Yokoyama, On almost global existence and local well posedness for some 3-D quasi-linear wave equations, Adv. Differential Equations 17 2012, 3–4, 267–306] and some new ingredients. Then, together with some weighted Sobolev inequalities, this estimate is used to show a refined version of almost global existence of classical solutions for nonlinear elastic waves with small initial data. Compared with former almost global existence results for nonlinear elastic waves due to John [F. John, Almost global existence of elastic waves of finite amplitude arising from small initial disturbances, Comm. Pure Appl. Math. 41 1988, 5, 615–666] and Klainerman and Sideris [S. Klainerman and T. C. Sideris, On almost global existence for nonrelativistic wave equations in 3D, Comm. Pure Appl. Math. 49 1996, 307–321], the main innovation of our result is that it considerably improves the amount of regularity of initial data, i.e., the Sobolev regularity of initial data is assumed to be the smallest among all the admissible Sobolev spaces of integer order in the standard local existence theory. Finally, in the radially symmetric case, we establish the almost global existence of a low regularity solution for every small initial data in H3×H2.

Abstract

In this paper, we analyze the embedding cell method, an algorithm which has been developed for the numerical homogenization of metal-ceramic composite materials. We show the convergence of the iteration scheme of this algorithm and the coincidence of the material properties predicted by the limit with the effective material properties provided by the analytical homogenization theory in two situations, namely for a one-dimensional linear elasticity model and a simple one-dimensional plasticity model.

Abstract

We give a presentation of the mathematical and numerical treatment of plate dynamics problems including rotational inertia. The presence of rotational inertia in the equation of motion makes the study of such problems interesting. We employ HCT finite elements for space discretization and the Newmark method for time discretization in FreeFEM++, and test such methods in some significant cases: a circular plate clamped all over its lateral surface, a rectangular plate simply supported all over its lateral surface, and an L-shaped clamped plate.

Abstract

We consider a hydrodynamic multi-phase field problem to model the interaction of deformable objects. The numerical approach considers one phase field variable for each object and allows for an independent adaptive mesh refinement for each variable. Using the special structure of various terms allows interpolating the solution on one mesh onto another without loss of information. Together with a general multi-mesh concept for the other terms speedup by a factor of two can be demonstrated which improves with the number of interacting objects. The general concept is demonstrated on an example describing the interaction of red blood cells in an idealized vessel.

Abstract

We prove the existence of weak solutions to the homogeneous wave equation on a suitable class of time-dependent domains. Using the approach suggested by De Giorgi and developed by Serra and Tilli, such solutions are approximated by minimizers of suitable functionals in space-time.

Abstract

We study the Γ-limit of Ambrosio–Tortorelli-type functionals Dε(u,v), whose dependence on the symmetrised gradient e(u) is different in 𝔸u and in e(u)-𝔸u, for a -elliptic symmetric operator 𝔸, in terms of the prefactor depending on the phase-field variable v. The limit energy depends both on the opening and on the surface of the crack, and is intermediate between the Griffith brittle fracture energy and the one considered by Focardi and Iurlano [Asymptotic analysis of Ambrosio–Tortorelli energies in linearized elasticity, SIAM J. Math. Anal. 46 2014, 4, 2936–2955]. In particular, we prove that G(S)BD functions with bounded 𝔸-variation are (S)BD.

Abstract

For the fractional diffusion-wave equation with the Caputo-Djrbashian fractional derivative of order α ∈ (1, 2) with respect to the time variable, we prove an analog of the principle of limiting amplitude (well-known for the wave equation and some other hyperbolic equations) and a pointwise stabilization property of solutions (similar to a well-known property of the heat equation and some other parabolic equations).

Abstract

In this work we study the continuity of four different notions of asymptotic behavior for a family of non-autonomous non-classical parabolic equations given by

{utγ(t)ΔutΔu=gϵ(t,u), in Ωu=0, on Ω.

in a smooth bounded domain Ω ⊂ ℝn, n ⩾ 3, where the terms gε are a small perturbation, in some sense, of a function f that depends only on u.